1. Watermarking of SAT Using Combinatorial Isolation Lemmas
Rupak Majumdar
EECS Dept. University of California, Berkeley, CA
Jennifer L. Wong
CS Dept. University of California, Los Angeles, CA
DAC, June, 2001

2. Watermarking Embedding of Information for ID or Proof of Authorship
Technique Adds Extra Constraints to the Problem
Effective Intellectual Property Protection Technique

3. Boolean Satisfiability
Instance: A set of variables V and a collection C of clauses over V.
Solution: A truth assignment for V such that at least one variable in each clause evaluates to true.
V = {v1, v2, v3}
C = {{v1, v2}, {v1’}, {v1’, v3},
{v1’, v2’, v3’}, {v3}}
Solution: v1 = False
v2 = True
v3 = True
V = {v1, v2, v3}
C = {{v1, v2}, {v1’}, {v1’, v3},
{v1’, v2’, v3’}, {v3}}

4. Boolean Satisfiability
First NP-complete Problem
Applications
Deterministic Test Pattern Generation
Delay Fault Testing
Logic Verification/Synthesis
FPGA Routing
AI, Operations Research, Combinatorial Optimization
Backtrack Search, Local Search, Algebraic Manipulation, Recursive Learning, …
Watermarking Techniques
Constraint-Based (Kahng ’98, Qu ‘99)

5. Fairness & Watermarking
All possible watermarking signatures of a given length result in a similar solutions space
Difficulty of finding a solution after WM (given length) -> Runtime
Quantify by # solutions
Solution Space

6. Fairness & Watermarking
V = {v1, v2, v3, v4}
C = { (v1 + v3 + v4’) (v2 + v4) (v1’+ v2+ v3’+ v4) (v2’+v3+v4’) }
Embed a Signature of 4 bits
Ave Solution Distance 0.85 & Ave Variance 0.21
Fair Technique

7. Credibility & Watermarking
# solutions after WM
# solutions of with quality  threshold
Effort required to find a particular solution

8. Credibility & Watermarking
V = {v1, v2, v3, v4}
C = { (v1 + v3 + v4’) (v2 + v4) (v1’+ v2+ v3’+ v4) (v2’+v3+v4’) }
Embed Signatures which are Multiples of Length 4

9. Watermarking Flow

10. Isolation Lemma (Valiant & Vazirani)
If f is any CNF formula in x1, x2, …, xn and w1, …wk {0,1}n, then one can construct in linear time a formula fk’ whose satisfying assignments v satisfy f and the equations v•w1 = v•w2 = …= v•wk = 0. Furthermore, one can construct a polynomial-size CNF formula fk in variables x1, x2, …, xn,y1, …, ym for some m s.t. there is a bijection between solutions of fk and fk’ defined by equality on the values of x1, x2, …, xn.
“NP is as easy as detecting unique solutions”
Isolates a solution by randomized reduction

11. Unique and Fair Solutions to SAT
CNF Formula over k Variables
Add Multiple Watermarks of length k
Adding Additional Constraints to the Formula
Let a  b  c denote the CNF formula
(a’ V b’ V c’) (a’ V b V c) (a V b V c’) (a V b’ V c)

12. Unique and Fair Solutions to SAT
f is a CNF formula
vi is the ith variable in the set V
wj is the jth watermark of k-bits
xy is the yth created variable
f *= f  ( vi1  vi2  …  vij  1)

13. Unique and Fair Solutions to SAT
V = {v1, v2, v3, v4}
f = { (v1,v2’, v3} {v1’, v3, v4} {v2,v3’,v4} }
V = {v1, v2, v3, v4}
f = { {v1,v2’, v3} {v1’, v3, v4} {v2,v3’,v4} }

14. Unique and Fair Solutions to SAT
V = {v1, v2, v3, v4}
f = { {v1,v2’, v3} {v1’, v3, v4} {v2,v3’,v4} } v1 v2 v3 v4 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

15. Unique and Fair Solutions to SAT
V = {v1, v2, v3, v4}
f = { {v1,v2’, v3} {v1’, v3, v4} {v2,v3’,v4} }
Watermark = 0101
Positions of 1’s = {2, 4}
f *= f  ( v2  v4  1)

16. Unique and Fair Solutions to SAT
f *= f  ( v2  v4  1)
( v2  v4  1) = (x1  v2  v4 )  (x1  1)
Create x1
V* = {v1, v2, v3, v4, x1}
(x1  v2  v4) (x1  1)
{x1’,v2’, v4’} {x1’,v2, v4}
{x1,v2’, v4} {x1,v2, v4’}
{x2’}

17. Unique and Fair Solutions to SAT
f **= { {v1,v2’,v3} {v1’,v3,v4} {v2,v3’,v4} {x1’,v2’,v4’}
{x1’,v2,v4} {x1,v2’,v4} {x1,v2,v4’} {x1’}} V1 V2 V3 V4
X1  V2  V4
X1  1 1

18. Unique and Fair Solutions to SAT
V* = {v1, v2, v3, v4, x1, x2}
f *= { {v1,v2’,v3} {v1’,v3,v4} {v2,v3’,v4} {x1’,v2’,v4’}
{x1’,v2,v4} {x1,v2’,v4} {x1,v2,v4’} {x1’}}
Watermark = 0011
Positions of 1’s = {3, 4}
f **= f *  ( v3  v4  1)

19. Unique and Fair Solutions to SAT
f **= f *  ( v3  v4  1)
( v3  v4  1) = (x2  v3  v4 )  (x2  1)
Create x2
V** = {v1, v2, v3, v4, x1, x2}
x2  v3  v4 x2  1
{x2’,v3’, v4’} {x2’,v3, v4}
{x2,v3’, v4} {x2,v3, v4’}
{x2’}

20. Unique and Fair Solutions to SAT
f **= { {v1,v2’,v3} {v1’,v3,v4} {v2,v3’,v4} {x1’,v2’,v4’} {x1’,v2,v4} {x1,v2’,v4} {x1,v2,v4’} {x1’} {x2’,v3’, v4’} {x2’,v3, v4} {x2,v3’, v4} {x2,v3, v4’} {x2’} } V1 V2 V3 V4
X1 
V2  V4
X1  1
X 2
V3  V4
X2  1 1

21. Experimentation Enviroment
SAT Instances
DIMACS Instances
Created Instance with Known # Solutions
Public Domain SAT Solvers
WalkSAT (Selman & Kautz ’94)
GSAT (Selman & Kautz ’92)
NTAB (Crawford & Auton ‘93)
Rel_SAT (Bayardo & Schrag ‘97)

22. Credibility Created Instances

23. Credibility (trade-off strength & runtime) DIMACS

24. Fairness (Runtime) DIMACS

25. Fairness (Runtime) Created Instances

26. Conclusion Ultimate Fairness and Credibility
Arbitrary Problem Application
Connection between Watermarking & Sound Mathematics & Theoretical Computer Science